Tiling Solver#
Tiling a n-dimensional polyomino with n-dimensional polyominoes.
This module defines two classes:
sage.combinat.tiling.Polyomino
class, to represent polyominoes in arbitrary dimension. The goal of this class is to return all the rotated, reflected and/or translated copies of a polyomino that are contained in a certain box.sage.combinat.tiling.TilingSolver
class, to solve the problem of tiling a \(n\)-dimensional polyomino with a set of \(n\)-dimensional polyominoes. One can specify if rotations and reflections are allowed or not and if pieces can be reused or not. This class convert the tiling data into rows of a matrix that are passed to the DLX solver. It also allows to compute the number of solutions.
This uses dancing links code which is in Sage. Dancing links were originally introduced by Donald Knuth in 2000 [Knuth1]. Knuth used dancing links to solve tilings of a region by 2d pentaminoes. Here we extend the method to any dimension.
In particular, the sage.games.quantumino
module is based on
the Tiling Solver and allows to solve the 3d Quantumino puzzle.
AUTHOR:
Sébastien Labbé, June 2011, initial version
Sébastien Labbé, July 2015, count solutions up to rotations
Sébastien Labbé, April 2017, tiling a polyomino, not only a rectangular box
EXAMPLES:
2d Easy Example#
Here is a 2d example. Let us try to fill the \(3 \times 2\) rectangle with a \(1 \times 2\) rectangle and a \(2 \times 2\) square. Obviously, there are two solutions:
sage: from sage.combinat.tiling import TilingSolver, Polyomino
sage: p = Polyomino([(0,0), (0,1)])
sage: q = Polyomino([(0,0), (0,1), (1,0), (1,1)])
sage: T = TilingSolver([p,q], box=[3,2])
sage: it = T.solve()
sage: next(it)
[Polyomino: [(0, 0), (0, 1), (1, 0), (1, 1)], Color: gray, Polyomino: [(2, 0), (2, 1)], Color: gray]
sage: next(it)
[Polyomino: [(1, 0), (1, 1), (2, 0), (2, 1)], Color: gray, Polyomino: [(0, 0), (0, 1)], Color: gray]
sage: next(it)
Traceback (most recent call last):
...
StopIteration
sage: T.number_of_solutions()
2
Scott’s pentamino problem#
As mentioned in the introduction of [Knuth1], Scott’s pentamino problem consists in tiling a chessboard leaving the center four squares vacant with the 12 distinct pentaminoes.
The 12 pentaminoes:
sage: from sage.combinat.tiling import Polyomino
sage: I = Polyomino([(0,0),(1,0),(2,0),(3,0),(4,0)], color='brown')
sage: N = Polyomino([(1,0),(1,1),(1,2),(0,2),(0,3)], color='yellow')
sage: L = Polyomino([(0,0),(1,0),(0,1),(0,2),(0,3)], color='magenta')
sage: U = Polyomino([(0,0),(1,0),(0,1),(0,2),(1,2)], color='violet')
sage: X = Polyomino([(1,0),(0,1),(1,1),(1,2),(2,1)], color='pink')
sage: W = Polyomino([(2,0),(2,1),(1,1),(1,2),(0,2)], color='green')
sage: P = Polyomino([(1,0),(2,0),(0,1),(1,1),(2,1)], color='orange')
sage: F = Polyomino([(1,0),(1,1),(0,1),(2,1),(2,2)], color='gray')
sage: Z = Polyomino([(0,0),(1,0),(1,1),(1,2),(2,2)], color='yellow')
sage: T = Polyomino([(0,0),(0,1),(1,1),(2,1),(0,2)], color='red')
sage: Y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)], color='green')
sage: V = Polyomino([(0,0),(0,1),(0,2),(1,0),(2,0)], color='blue')
A \(8 \times 8\) chessboard leaving the center four squares vacant:
sage: import itertools
sage: s = set(itertools.product(range(8), repeat=2))
sage: s.difference_update([(3,3), (3,4), (4,3), (4,4)])
sage: chessboard = Polyomino(s)
sage: len(chessboard)
60
This problem is represented by a matrix made of 1568 rows and 72 columns. It has 65 different solutions up to isometries:
sage: from sage.combinat.tiling import TilingSolver
sage: T = TilingSolver([I,N,L,U,X,W,P,F,Z,T,Y,V], box=chessboard, reflection=True)
sage: T
Tiling solver of 12 pieces into a box of size 60
Rotation allowed: True
Reflection allowed: True
Reusing pieces allowed: False
sage: len(T.rows()) # long time
1568
sage: T.number_of_solutions() # long time
520
sage: 520 / 8
65
Showing one solution:
sage: solution = next(T.solve()) # long time
sage: G = sum([piece.show2d() for piece in solution], Graphics()) # long time # optional - sage.plot
sage: G.show(aspect_ratio=1, axes=False) # long time # optional - sage.plot
1d Easy Example#
Here is an easy one dimensional example where we try to tile a stick of length 6 with three sticks of length 1, 2 and 3. There are six solutions:
sage: p = Polyomino([[0]])
sage: q = Polyomino([[0],[1]])
sage: r = Polyomino([[0],[1],[2]])
sage: T = TilingSolver([p,q,r], box=[6])
sage: len(T.rows())
15
sage: it = T.solve()
sage: next(it)
[Polyomino: [(0)], Color: gray, Polyomino: [(1), (2)], Color: gray, Polyomino: [(3), (4), (5)], Color: gray]
sage: next(it)
[Polyomino: [(0)], Color: gray, Polyomino: [(1), (2), (3)], Color: gray, Polyomino: [(4), (5)], Color: gray]
sage: T.number_of_solutions()
6
2d Puzzle allowing reflections#
The following is a puzzle owned by Florent Hivert:
sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: L = []
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)], 'yellow'))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2)], "black"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,3)], "gray"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,3)],"cyan"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,0),(1,1)],"red"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,1),(1,2)],"blue"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(0,3),(1,1),(1,3)],"green"))
sage: L.append(Polyomino([(0,1),(0,2),(0,3),(1,0),(1,1),(1,3)],"magenta"))
sage: L.append(Polyomino([(0,1),(0,2),(0,3),(1,0),(1,1),(1,2)],"orange"))
sage: L.append(Polyomino([(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)],"pink"))
By default, rotations are allowed and reflections are not. In this case, there are no solution for tiling a \(8 \times 8\) rectangular box:
sage: T = TilingSolver(L, box=(8,8))
sage: T.number_of_solutions() # long time (2.5 s)
0
If reflections are allowed, there are solutions. Solve the puzzle and show one solution:
sage: T = TilingSolver(L, box=(8,8), reflection=True)
sage: solution = next(T.solve()) # long time (7s)
sage: G = sum([piece.show2d() for piece in solution], Graphics()) # long time (<1s) # optional - sage.plot
sage: G.show(aspect_ratio=1, axes=False) # long time (2s) # optional - sage.plot
Compute the number of solutions:
sage: T.number_of_solutions() # long time (2.6s)
328
Create a animation of all the solutions:
sage: a = T.animate() # not tested
sage: a # not tested
Animation with 328 frames
3d Puzzle#
The same thing done in 3d without allowing reflections this time:
sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: L = []
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,2,0),(1,3,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,2,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,3,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,3,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,1,0),(1,2,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(0,3,0),(1,1,0),(1,3,0)]))
sage: L.append(Polyomino([(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,3,0)]))
sage: L.append(Polyomino([(0,1,0),(0,2,0),(0,3,0),(1,0,0),(1,1,0),(1,2,0)]))
sage: L.append(Polyomino([(0,0,0),(0,1,0),(0,2,0),(1,0,0),(1,1,0),(1,2,0)]))
Solve the puzzle and show one solution:
sage: T = TilingSolver(L, box=(8,8,1))
sage: solution = next(T.solve()) # long time (8s)
sage: G = sum([p.show3d(size=0.85) for p in solution], Graphics()) # long time (<1s)
sage: G.show(aspect_ratio=1, viewer='tachyon') # long time (2s)
Let us compute the number of solutions:
sage: T.number_of_solutions() # long time (3s)
328
Donald Knuth example : the Y pentamino#
Donald Knuth [Knuth1] considered the problem of packing 45 Y pentaminoes into a \(15 \times 15\) square:
sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)])
sage: T = TilingSolver([y], box=(5,10), reusable=True, reflection=True)
sage: T.number_of_solutions()
10
sage: solution = next(T.solve())
sage: G = sum([p.show2d() for p in solution], Graphics()) # optional - sage.plot
sage: G.show(aspect_ratio=1) # long time (2s) # optional - sage.plot
sage: T = TilingSolver([y], box=(15,15), reusable=True, reflection=True)
sage: T.number_of_solutions() # not tested
1696
Up to the symmetries of the square, there are 212 distinct solutions:
sage: 1696 // 8
212
Animation of Donald Knuth’s dancing links#
Animation of the solutions:
sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: Y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)], color='yellow')
sage: T = TilingSolver([Y], box=(15,15), reusable=True, reflection=True)
sage: a = T.animate(stop=40); a # long time # optional -- ImageMagick sage.plot
Animation with 40 frames
Incremental animation of the solutions (one piece is removed/added at a time):
sage: a = T.animate('incremental', stop=40) # long time # optional -- ImageMagick sage.plot
sage: a # long time # optional -- ImageMagick sage.plot
Animation with 40 frames
sage: a.show(delay=50, iterations=1) # long time # optional -- ImageMagick sage.plot
5d Easy Example#
Here is a 5d example. Let us try to fill the \(2 \times 2 \times 2 \times 2 \times 2\) rectangle with reusable \(1 \times 1 \times 1 \times 1 \times 1\) rectangles. Obviously, there is one solution:
sage: from sage.combinat.tiling import Polyomino, TilingSolver
sage: p = Polyomino([(0,0,0,0,0)])
sage: T = TilingSolver([p], box=(2,2,2,2,2), reusable=True)
sage: rows = T.rows() # long time (3s)
sage: rows # long time (fast)
[[0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]]
sage: T.number_of_solutions() # long time (fast)
1
REFERENCES:
- class sage.combinat.tiling.Polyomino(coords, color='gray', dimension=None)#
Bases:
SageObject
A polyomino in \(\ZZ^d\).
The polyomino is the union of the unit square (or cube, or n-cube) centered at those coordinates. Such an object should be connected, but the code does not make this assumption.
INPUT:
coords
– iterable of integer coordinates in \(\ZZ^d\)color
– string (default:'gray'
), color for displaydimension
– integer (default:None
), dimension of the space, ifNone
, it is guessed from thecoords
ifcoords
is non empty
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue') Polyomino: [(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)], Color: blue
- boundary()#
Return the boundary of a 2d polyomino.
INPUT:
self
- a 2d polyomino
OUTPUT:
list of edges (an edge is a pair of adjacent 2d coordinates)
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0), (1,0), (0,1), (1,1)]) sage: sorted(p.boundary()) [((-0.5, -0.5), (-0.5, 0.5)), ((-0.5, -0.5), (0.5, -0.5)), ((-0.5, 0.5), (-0.5, 1.5)), ((-0.5, 1.5), (0.5, 1.5)), ((0.5, -0.5), (1.5, -0.5)), ((0.5, 1.5), (1.5, 1.5)), ((1.5, -0.5), (1.5, 0.5)), ((1.5, 0.5), (1.5, 1.5))] sage: len(_) 8 sage: p = Polyomino([(5,5)]) sage: sorted(p.boundary()) [((4.5, 4.5), (4.5, 5.5)), ((4.5, 4.5), (5.5, 4.5)), ((4.5, 5.5), (5.5, 5.5)), ((5.5, 4.5), (5.5, 5.5))]
- bounding_box()#
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink') sage: p.bounding_box() [[0, 0, 0], [1, 2, 1]]
- canonical()#
Return the translated copy of
self
having minimal and nonnegative coordinatesEXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink') sage: p Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink sage: p.canonical() Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink
- canonical_isometric_copies(orientation_preserving=True, mod_box_isometries=False)#
Return the list of image of
self
under isometries of the \(n\)-cube where the coordinates are all nonnegative and minimal.INPUT:
orientation_preserving
– bool (optional, default:True
); ifTrue
, the group of isometries of the \(n\)-cube is restricted to those that preserve the orientation, i.e. of determinant 1.mod_box_isometries
– bool (default:False
), whether to quotient the group of isometries of the \(n\)-cube by the subgroup of isometries of the \(a_1\times a_2\cdots \times a_n\) rectangular box where are the \(a_i\) are assumed to be distinct.
OUTPUT:
set of Polyomino
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue') sage: s = p.canonical_isometric_copies() sage: len(s) 12
With the non orientation-preserving:
sage: s = p.canonical_isometric_copies(orientation_preserving=False) sage: len(s) 24
Modulo rotation by angle 180 degrees:
sage: s = p.canonical_isometric_copies(mod_box_isometries=True) sage: len(s) 3
- center()#
Return the center of the polyomino.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0),(0,0,1)]) sage: p.center() (0, 0, 1/2)
In 3d:
sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink') sage: p.center() (4/5, 4/5, 1/5)
In 2d:
sage: p = Polyomino([(0,0),(1,0),(1,1),(1,2)]) sage: p.center() (3/4, 3/4)
- color(color=None)#
Return or change the color of the polyomino.
INPUT:
color
– string, RBG tuple orNone
(default:None
), ifNone
, it returns the current color
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue') sage: p.color() 'blue'
- frozenset()#
Return the elements of \(\ZZ^d\) in the polyomino as a frozenset.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='red') sage: p.frozenset() frozenset({(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)})
- intersection(other)#
Return the intersection of
self
andother
.INPUT:
other
- a polyomino
OUTPUT:
polyomino
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: a = Polyomino([(0,0)]) sage: b = Polyomino([(0,0), (0,1), (1,1), (2,1)]) sage: a.intersection(b) Polyomino: [(0, 0)], Color: gray sage: a.intersection(b+(1,1)) Polyomino: [], Color: gray
- isometric_copies(box, orientation_preserving=True, mod_box_isometries=False)#
Return the translated and isometric images of
self
that lies in the box.INPUT:
box
– Polyomino or tuple of integers (size of a box)orientation_preserving
– bool (optional, default:True
); IfTrue
, the group of isometries of the \(n\)-cube is restricted to those that preserve the orientation, i.e. of determinant 1.mod_box_isometries
– bool (default:False
), whether to quotient the group of isometries of the \(n\)-cube by the subgroup of isometries of the \(a_1\times a_2\cdots \times a_n\) rectangular box where are the \(a_i\) are assumed to be distinct.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink') sage: L = list(p.isometric_copies(box=(5,8,2))) sage: len(L) 360
sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,2,0),(1,2,1)], color='orange') sage: L = list(p.isometric_copies(box=(5,8,2))) sage: len(L) 180 sage: L = list(p.isometric_copies((5,8,2), False)) sage: len(L) 360 sage: L = list(p.isometric_copies((5,8,2), mod_box_isometries=True)) sage: len(L) 45
sage: p = Polyomino([(0,0), (1,0), (0,1)]) sage: b = Polyomino([(0,0), (1,0), (2,0), (0,1), (1,1), (0,2)]) sage: sorted(p.isometric_copies(b), key=lambda p: p.sorted_list()) [Polyomino: [(0, 0), (0, 1), (1, 0)], Color: gray, Polyomino: [(0, 0), (0, 1), (1, 1)], Color: gray, Polyomino: [(0, 0), (1, 0), (1, 1)], Color: gray, Polyomino: [(0, 1), (0, 2), (1, 1)], Color: gray, Polyomino: [(0, 1), (1, 0), (1, 1)], Color: gray, Polyomino: [(1, 0), (1, 1), (2, 0)], Color: gray]
- isometric_copies_intersection(box, orientation_preserving=True)#
Return the set of non empty intersections of isometric images of
self
with a polyomino.INPUT:
box
– Polyomino or tuple of integers (size of a box)orientation_preserving
– bool (optional, default:True
); ifTrue
, the group of isometries of the \(n\)-cube is restricted to those that preserve the orientation, i.e. of determinant 1.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0),(1,0)], color='deeppink') sage: sorted(sorted(a.frozenset()) for a in p.isometric_copies_intersection(box=(2,3))) [[(0, 0)], [(0, 0), (0, 1)], [(0, 0), (1, 0)], [(0, 1)], [(0, 1), (0, 2)], [(0, 1), (1, 1)], [(0, 2)], [(0, 2), (1, 2)], [(1, 0)], [(1, 0), (1, 1)], [(1, 1)], [(1, 1), (1, 2)], [(1, 2)]]
- neighbor_edges()#
Return an iterator over the pairs of neighbor coordinates inside of the polyomino.
Two points \(P\) and \(Q\) in the polyomino are neighbor if \(P - Q\) has one coordinate equal to \(+1\) or \(-1\) and zero everywhere else.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0),(0,0,1)]) sage: [sorted(edge) for edge in p.neighbor_edges()] [[(0, 0, 0), (0, 0, 1)]]
In 3d:
sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink') sage: L = sorted(sorted(edge) for edge in p.neighbor_edges()) sage: for a in L: a [(0, 0, 0), (1, 0, 0)] [(1, 0, 0), (1, 1, 0)] [(1, 1, 0), (1, 1, 1)] [(1, 1, 0), (1, 2, 0)]
In 2d:
sage: p = Polyomino([(0,0),(1,0),(1,1),(1,2)]) sage: L = sorted(sorted(edge) for edge in p.neighbor_edges()) sage: for a in L: a [(0, 0), (1, 0)] [(1, 0), (1, 1)] [(1, 1), (1, 2)]
- self_surrounding(radius, remove_incomplete_copies=True, ncpus=None)#
Return a list of isometric copies of
self
surrounding it with an annulus of given radius.INPUT:
self
- a polyomino of dimension 2radius
- integerremove_incomplete_copies
– bool (default:True
), whether to keep only complete copies ofself
in the outputncpus
– integer (default:None
), maximal number of subprocesses to use at the same time. IfNone
, it detects the number of effective CPUs in the system usingsage.parallel.ncpus.ncpus()
. Ifncpus=1
, the first solution is searched serially.
OUTPUT:
list of polyominoes
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: H = Polyomino([(-1, 1), (-1, 4), (-1, 7), (0, 0), (0, 1), (0, 2), ....: (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (1, 1), (1, 2), ....: (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (2, 0), (2, 2), ....: (2, 3), (2, 5), (2, 6), (2, 8)]) sage: solution = H.self_surrounding(8) sage: G = sum([p.show2d() for p in solution], Graphics()) # optional - sage.plot
sage: solution = H.self_surrounding(8, remove_incomplete_copies=False) sage: G = sum([p.show2d() for p in solution], Graphics()) # optional - sage.plot
- show2d(size=0.7, color='black', thickness=1)#
Return a 2d Graphic object representing the polyomino.
INPUT:
self
- a polyomino of dimension 2size
- number (optional, default:0.7
), the size of each square.color
- color (optional, default:'black'
), color of the boundary line.thickness
- number (optional, default:1
), how thick the boundary line is.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0),(1,0),(1,1),(1,2)], color='deeppink') sage: p.show2d() # long time (0.5s) # optional -- sage.plot Graphics object consisting of 17 graphics primitives
- show3d(size=1)#
Return a 3d Graphic object representing the polyomino.
INPUT:
self
- a polyomino of dimension 3size
- number (optional, default:1
), the size of each1 \times 1 \times 1
cube. This does a homothety with respect to the center of the polyomino.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue') sage: p.show3d() # long time (2s) # optional -- sage.plot Graphics3d Object
- sorted_list()#
Return the color of the polyomino.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0), (0,1,0), (1,1,0), (1,1,1)], color='blue') sage: p.sorted_list() [(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 1, 1)]
- translated_copies(box)#
Return an iterator over the translated images of
self
inside a polyomino.INPUT:
box
– Polyomino or tuple of integers (size of a box)
OUTPUT:
iterator of 3d polyominoes
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0,0),(1,0,0),(1,1,0),(1,1,1),(1,2,0)], color='deeppink') sage: for t in p.translated_copies(box=(5,8,2)): t Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink Polyomino: [(0, 1, 0), (1, 1, 0), (1, 2, 0), (1, 2, 1), (1, 3, 0)], Color: deeppink Polyomino: [(0, 2, 0), (1, 2, 0), (1, 3, 0), (1, 3, 1), (1, 4, 0)], Color: deeppink Polyomino: [(0, 3, 0), (1, 3, 0), (1, 4, 0), (1, 4, 1), (1, 5, 0)], Color: deeppink Polyomino: [(0, 4, 0), (1, 4, 0), (1, 5, 0), (1, 5, 1), (1, 6, 0)], Color: deeppink Polyomino: [(0, 5, 0), (1, 5, 0), (1, 6, 0), (1, 6, 1), (1, 7, 0)], Color: deeppink Polyomino: [(1, 0, 0), (2, 0, 0), (2, 1, 0), (2, 1, 1), (2, 2, 0)], Color: deeppink Polyomino: [(1, 1, 0), (2, 1, 0), (2, 2, 0), (2, 2, 1), (2, 3, 0)], Color: deeppink Polyomino: [(1, 2, 0), (2, 2, 0), (2, 3, 0), (2, 3, 1), (2, 4, 0)], Color: deeppink Polyomino: [(1, 3, 0), (2, 3, 0), (2, 4, 0), (2, 4, 1), (2, 5, 0)], Color: deeppink Polyomino: [(1, 4, 0), (2, 4, 0), (2, 5, 0), (2, 5, 1), (2, 6, 0)], Color: deeppink Polyomino: [(1, 5, 0), (2, 5, 0), (2, 6, 0), (2, 6, 1), (2, 7, 0)], Color: deeppink Polyomino: [(2, 0, 0), (3, 0, 0), (3, 1, 0), (3, 1, 1), (3, 2, 0)], Color: deeppink Polyomino: [(2, 1, 0), (3, 1, 0), (3, 2, 0), (3, 2, 1), (3, 3, 0)], Color: deeppink Polyomino: [(2, 2, 0), (3, 2, 0), (3, 3, 0), (3, 3, 1), (3, 4, 0)], Color: deeppink Polyomino: [(2, 3, 0), (3, 3, 0), (3, 4, 0), (3, 4, 1), (3, 5, 0)], Color: deeppink Polyomino: [(2, 4, 0), (3, 4, 0), (3, 5, 0), (3, 5, 1), (3, 6, 0)], Color: deeppink Polyomino: [(2, 5, 0), (3, 5, 0), (3, 6, 0), (3, 6, 1), (3, 7, 0)], Color: deeppink Polyomino: [(3, 0, 0), (4, 0, 0), (4, 1, 0), (4, 1, 1), (4, 2, 0)], Color: deeppink Polyomino: [(3, 1, 0), (4, 1, 0), (4, 2, 0), (4, 2, 1), (4, 3, 0)], Color: deeppink Polyomino: [(3, 2, 0), (4, 2, 0), (4, 3, 0), (4, 3, 1), (4, 4, 0)], Color: deeppink Polyomino: [(3, 3, 0), (4, 3, 0), (4, 4, 0), (4, 4, 1), (4, 5, 0)], Color: deeppink Polyomino: [(3, 4, 0), (4, 4, 0), (4, 5, 0), (4, 5, 1), (4, 6, 0)], Color: deeppink Polyomino: [(3, 5, 0), (4, 5, 0), (4, 6, 0), (4, 6, 1), (4, 7, 0)], Color: deeppink
This method is independent of the translation of the polyomino:
sage: q = Polyomino([(0,0,0), (1,0,0)]) sage: list(q.translated_copies((2,2,1))) [Polyomino: [(0, 0, 0), (1, 0, 0)], Color: gray, Polyomino: [(0, 1, 0), (1, 1, 0)], Color: gray] sage: q = Polyomino([(34,7,-9), (35,7,-9)]) sage: list(q.translated_copies((2,2,1))) [Polyomino: [(0, 0, 0), (1, 0, 0)], Color: gray, Polyomino: [(0, 1, 0), (1, 1, 0)], Color: gray]
Inside smaller boxes:
sage: list(p.translated_copies(box=(2,2,3))) [] sage: list(p.translated_copies(box=(2,3,2))) [Polyomino: [(0, 0, 0), (1, 0, 0), (1, 1, 0), (1, 1, 1), (1, 2, 0)], Color: deeppink] sage: list(p.translated_copies(box=(3,2,2))) [] sage: list(p.translated_copies(box=(1,1,1))) []
Using a Polyomino as input:
sage: b = Polyomino([(0,0), (0,1), (0,2), (1,0), (1,1), (1,2)]) sage: p = Polyomino([(0,0)]) sage: list(p.translated_copies(b)) [Polyomino: [(0, 0)], Color: gray, Polyomino: [(0, 1)], Color: gray, Polyomino: [(0, 2)], Color: gray, Polyomino: [(1, 0)], Color: gray, Polyomino: [(1, 1)], Color: gray, Polyomino: [(1, 2)], Color: gray]
sage: p = Polyomino([(0,0), (1,0), (0,1)]) sage: b = Polyomino([(0,0), (1,0), (2,0), (0,1), (1,1), (0,2)]) sage: list(p.translated_copies(b)) [Polyomino: [(0, 0), (0, 1), (1, 0)], Color: gray, Polyomino: [(0, 1), (0, 2), (1, 1)], Color: gray, Polyomino: [(1, 0), (1, 1), (2, 0)], Color: gray]
- translated_copies_intersection(box)#
Return the set of non empty intersections of translated images of
self
with a polyomino.INPUT:
box
– Polyomino or tuple of integers (size of a box)
OUTPUT:
set of 3d polyominoes
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino sage: p = Polyomino([(0,0),(1,0)], color='deeppink') sage: sorted(sorted(a.frozenset()) for a in p.translated_copies_intersection(box=(2,3))) [[(0, 0)], [(0, 0), (1, 0)], [(0, 1)], [(0, 1), (1, 1)], [(0, 2)], [(0, 2), (1, 2)], [(1, 0)], [(1, 1)], [(1, 2)]]
Using a Polyomino as input:
sage: b = Polyomino([(0,0), (0,1), (0,2), (1,0), (2,0)]) sage: p = Polyomino([(0,0), (1,0)]) sage: sorted(sorted(a.frozenset()) for a in p.translated_copies_intersection(b)) [[(0, 0)], [(0, 0), (1, 0)], [(0, 1)], [(0, 2)], [(1, 0), (2, 0)], [(2, 0)]]
- class sage.combinat.tiling.TilingSolver(pieces, box, rotation=True, reflection=False, reusable=False, outside=False)#
Bases:
SageObject
Tiling solver
Solve the problem of tiling a polyomino with a certain number of polyominoes.
INPUT:
pieces
– iterable of Polyominoesbox
– Polyomino or tuple of integers (size of a box)rotation
– bool (optional, default:True
), whether to allow rotationsreflection
– bool (optional, default:False
), whether to allow reflectionsreusable
– bool (optional, default:False
), whether to allow the pieces to be reusedoutside
– bool (optional, default:False
), whether to allow pieces to partially go outside of the box (all non-empty intersection of the pieces with the box are considered)
EXAMPLES:
By default, rotations are allowed and reflections are not allowed:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: T Tiling solver of 3 pieces into a box of size 6 Rotation allowed: True Reflection allowed: False Reusing pieces allowed: False
Solutions are given by an iterator:
sage: it = T.solve() sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: gray Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray
Another solution:
sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: gray Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray
Tiling of a polyomino by polyominoes:
sage: b = Polyomino([(0,0), (1,0), (1,1), (2,1), (1,2), (2,2), (0,3), (1,3)]) sage: p = Polyomino([(0,0), (1,0)]) sage: T = TilingSolver([p], box=b, reusable=True) sage: T.number_of_solutions() 2
- animate(partial=None, stop=None, size=0.75, axes=False)#
Return an animation of evolving solutions.
INPUT:
partial
- string (optional, default:None
), whether to include partial (incomplete) solutions. It can be one of the following:None
- include only complete solutions'common_prefix'
- common prefix between two consecutive solutions'incremental'
- one piece change at a time
stop
- integer (optional, default:None
), number of framessize
- number (optional, default:0.75
), the size of each1 \times 1
square. This does a homothety with respect to the center of each polyomino.axes
- bool (optional, default:False
), whether the x and y axes are shown.
EXAMPLES:
sage: from sage.combinat.tiling import Polyomino, TilingSolver sage: y = Polyomino([(0,0),(1,0),(2,0),(3,0),(2,1)], color='cyan') sage: T = TilingSolver([y], box=(5,10), reusable=True, reflection=True) sage: a = T.animate() # optional - sage.plot sage: a # optional -- ImageMagick # long time # optional - sage.plot Animation with 10 frames
Include partial solutions (common prefix between two consecutive solutions):
sage: a = T.animate('common_prefix') # optional - sage.plot sage: a # optional -- ImageMagick # long time # optional - sage.plot Animation with 19 frames
Incremental solutions (one piece removed or added at a time):
sage: a = T.animate('incremental') # long time (2s) # optional - sage.plot sage: a # long time (2s) # optional -- ImageMagick sage.plot Animation with 123 frames
sage: a.show() # optional -- ImageMagick # long time # optional - sage.plot
The
show
function takes arguments to specify the delay between frames (measured in hundredths of a second, default value 20) and the number of iterations (default value 0, which means to iterate forever). To iterate 4 times with half a second between each frame:sage: a.show(delay=50, iterations=4) # optional -- ImageMagick # long time # optional - sage.plot
Limit the number of frames:
sage: a = T.animate('incremental', stop=13) # not tested # optional - sage.plot sage: a # not tested # optional - sage.plot Animation with 13 frames
- coord_to_int_dict()#
Return a dictionary mapping coordinates to integers.
OUTPUT:
dict
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: A = T.coord_to_int_dict() sage: sorted(A.items()) [((0, 0, 0), 3), ((0, 0, 1), 4), ((0, 0, 2), 5), ((0, 0, 3), 6), ((0, 0, 4), 7), ((0, 0, 5), 8)]
Reusable pieces:
sage: p = Polyomino([(0,0), (0,1)]) sage: q = Polyomino([(0,0), (0,1), (1,0), (1,1)]) sage: T = TilingSolver([p,q], box=[3,2], reusable=True) sage: B = T.coord_to_int_dict() sage: sorted(B.items()) [((0, 0), 0), ((0, 1), 1), ((1, 0), 2), ((1, 1), 3), ((2, 0), 4), ((2, 1), 5)]
- dlx_solver()#
Return the sage DLX solver of that tiling problem.
OUTPUT:
DLX Solver
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: T.dlx_solver() Dancing links solver for 9 columns and 15 rows
- int_to_coord_dict()#
Return a dictionary mapping integers to coordinates.
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: B = T.int_to_coord_dict() sage: sorted(B.items()) [(3, (0, 0, 0)), (4, (0, 0, 1)), (5, (0, 0, 2)), (6, (0, 0, 3)), (7, (0, 0, 4)), (8, (0, 0, 5))]
Reusable pieces:
sage: from sage.combinat.tiling import Polyomino, TilingSolver sage: p = Polyomino([(0,0), (0,1)]) sage: q = Polyomino([(0,0), (0,1), (1,0), (1,1)]) sage: T = TilingSolver([p,q], box=[3,2], reusable=True) sage: B = T.int_to_coord_dict() sage: sorted(B.items()) [(0, (0, 0)), (1, (0, 1)), (2, (1, 0)), (3, (1, 1)), (4, (2, 0)), (5, (2, 1))]
- is_suitable()#
Return whether the volume of the box is equal to sum of the volume of the polyominoes and the number of rows sent to the DLX solver is larger than zero.
If these conditions are not verified, then the problem is not suitable in the sense that there are no solution.
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: T.is_suitable() True sage: T = TilingSolver([p,q,r], box=(1,1,7)) sage: T.is_suitable() False
- nrows_per_piece()#
Return the number of rows necessary by each piece.
OUTPUT:
list
EXAMPLES:
sage: from sage.games.quantumino import QuantuminoSolver sage: q = QuantuminoSolver(0) sage: T = q.tiling_solver() sage: T.nrows_per_piece() # long time (10s) [360, 360, 360, 360, 360, 180, 180, 672, 672, 360, 360, 180, 180, 360, 360, 180]
- number_of_solutions()#
Return the number of distinct solutions.
OUTPUT:
integer
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0)]) sage: q = Polyomino([(0,0), (0,1)]) sage: r = Polyomino([(0,0), (0,1), (0,2)]) sage: T = TilingSolver([p,q,r], box=(1,6)) sage: T.number_of_solutions() 6
sage: T = TilingSolver([p,q,r], box=(1,7)) sage: T.number_of_solutions() 0
- pieces()#
Return the list of pieces.
OUTPUT:
list of 3d polyominoes
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: for p in T._pieces: p Polyomino: [(0, 0, 0)], Color: gray Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray Polyomino: [(0, 0, 0), (0, 0, 1), (0, 0, 2)], Color: gray
- row_to_polyomino(row_number)#
Return a polyomino associated to a row.
INPUT:
row_number
– integer, the i-th row
OUTPUT:
polyomino
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: a = Polyomino([(0,0,0), (0,0,1), (1,0,0)], color='blue') sage: b = Polyomino([(0,0,0), (1,0,0), (0,1,0)], color='red') sage: T = TilingSolver([a,b], box=(2,1,3)) sage: len(T.rows()) 16
sage: T.row_to_polyomino(7) Polyomino: [(0, 0, 2), (1, 0, 1), (1, 0, 2)], Color: blue
sage: T.row_to_polyomino(13) Polyomino: [(0, 0, 1), (1, 0, 1), (1, 0, 2)], Color: red
- rows()#
Creation of the rows
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: rows = T.rows() sage: for row in rows: row [0, 3] [0, 4] [0, 5] [0, 6] [0, 7] [0, 8] [1, 3, 4] [1, 4, 5] [1, 5, 6] [1, 6, 7] [1, 7, 8] [2, 3, 4, 5] [2, 4, 5, 6] [2, 5, 6, 7] [2, 6, 7, 8]
- rows_for_piece(i, mod_box_isometries=False)#
Return the rows for the i-th piece.
INPUT:
i
– integer, the i-th piecemod_box_isometries
– bool (default:False
), whether to consider only rows for positions up to the action of the quotient the group of isometries of the \(n\)-cube by the subgroup of isometries of the \(a_1\times a_2\cdots \times a_n\) rectangular box where are the \(a_i\) are assumed to be distinct.
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: T.rows_for_piece(0) [[0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8]] sage: T.rows_for_piece(1) [[1, 3, 4], [1, 4, 5], [1, 5, 6], [1, 6, 7], [1, 7, 8]] sage: T.rows_for_piece(2) [[2, 3, 4, 5], [2, 4, 5, 6], [2, 5, 6, 7], [2, 6, 7, 8]]
Less rows when using
mod_box_isometries=True
:sage: a = Polyomino([(0,0,0), (0,0,1), (1,0,0)]) sage: b = Polyomino([(0,0,0), (1,0,0), (0,1,0)]) sage: T = TilingSolver([a,b], box=(2,1,3)) sage: T.rows_for_piece(0) [[0, 2, 3, 5], [0, 3, 4, 6], [0, 2, 3, 6], [0, 3, 4, 7], [0, 2, 5, 6], [0, 3, 6, 7], [0, 3, 5, 6], [0, 4, 6, 7]] sage: T.rows_for_piece(0, mod_box_isometries=True) [[0, 2, 3, 5], [0, 3, 4, 6]] sage: T.rows_for_piece(1, mod_box_isometries=True) [[1, 2, 3, 5], [1, 3, 4, 6]]
- solve(partial=None)#
Return an iterator of list of polyominoes that are an exact cover of the box.
INPUT:
partial
- string (optional, default:None
), whether to include partial (incomplete) solutions. It can be one of the following:None
- include only complete solution'common_prefix'
- common prefix between two consecutive solutions'incremental'
- one piece change at a time
OUTPUT:
iterator of list of polyominoes
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: it = T.solve() sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: gray Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: gray Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray sage: for p in next(it): p Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray Polyomino: [(0, 0, 2), (0, 0, 3), (0, 0, 4)], Color: gray Polyomino: [(0, 0, 5)], Color: gray
Including the partial solutions:
sage: it = T.solve(partial='common_prefix') sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: gray Polyomino: [(0, 0, 1), (0, 0, 2)], Color: gray Polyomino: [(0, 0, 3), (0, 0, 4), (0, 0, 5)], Color: gray sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: gray sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: gray Polyomino: [(0, 0, 1), (0, 0, 2), (0, 0, 3)], Color: gray Polyomino: [(0, 0, 4), (0, 0, 5)], Color: gray sage: for p in next(it): p sage: for p in next(it): p Polyomino: [(0, 0, 0), (0, 0, 1)], Color: gray Polyomino: [(0, 0, 2), (0, 0, 3), (0, 0, 4)], Color: gray Polyomino: [(0, 0, 5)], Color: gray
Colors are preserved when the polyomino can be reused:
sage: p = Polyomino([(0,0,0)], color='yellow') sage: q = Polyomino([(0,0,0), (0,0,1)], color='yellow') sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)], color='yellow') sage: T = TilingSolver([p,q,r], box=(1,1,6), reusable=True) sage: it = T.solve() sage: for p in next(it): p Polyomino: [(0, 0, 0)], Color: yellow Polyomino: [(0, 0, 1)], Color: yellow Polyomino: [(0, 0, 2)], Color: yellow Polyomino: [(0, 0, 3)], Color: yellow Polyomino: [(0, 0, 4)], Color: yellow Polyomino: [(0, 0, 5)], Color: yellow
- space()#
Return an iterator over all the non negative integer coordinates contained in the space to tile.
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: list(T.space()) [(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 0, 3), (0, 0, 4), (0, 0, 5)]
- starting_rows()#
Return the starting rows for each piece.
EXAMPLES:
sage: from sage.combinat.tiling import TilingSolver, Polyomino sage: p = Polyomino([(0,0,0)]) sage: q = Polyomino([(0,0,0), (0,0,1)]) sage: r = Polyomino([(0,0,0), (0,0,1), (0,0,2)]) sage: T = TilingSolver([p,q,r], box=(1,1,6)) sage: T.starting_rows() [0, 6, 11, 15]
- sage.combinat.tiling.ncube_isometry_group(n, orientation_preserving=True)#
Return the isometry group of the \(n\)-cube as a list of matrices.
INPUT:
n
– positive integer, dimension of the spaceorientation_preserving
– bool (optional, default:True
), whether the orientation is preserved
OUTPUT:
list of matrices
EXAMPLES:
sage: from sage.combinat.tiling import ncube_isometry_group sage: ncube_isometry_group(2) [ [1 0] [ 0 1] [-1 0] [ 0 -1] [0 1], [-1 0], [ 0 -1], [ 1 0] ] sage: ncube_isometry_group(2, orientation_preserving=False) [ [1 0] [ 0 -1] [ 1 0] [ 0 1] [0 1] [-1 0] [ 0 -1] [-1 0] [0 1], [-1 0], [ 0 -1], [-1 0], [1 0], [ 0 -1], [ 1 0], [ 0 1] ]
There are 24 orientation preserving isometries of the 3-cube:
sage: ncube_isometry_group(3) [ [1 0 0] [ 1 0 0] [ 1 0 0] [ 0 1 0] [0 1 0] [ 0 0 1] [0 1 0] [ 0 0 1] [ 0 0 -1] [-1 0 0] [0 0 1] [ 0 -1 0] [0 0 1], [ 0 -1 0], [ 0 1 0], [ 0 0 1], [1 0 0], [ 1 0 0], [-1 0 0] [ 0 -1 0] [-1 0 0] [-1 0 0] [ 0 -1 0] [ 0 0 -1] [ 0 -1 0] [ 0 0 -1] [ 0 0 -1] [ 0 1 0] [ 0 0 1] [ 1 0 0] [ 0 0 1], [ 1 0 0], [ 0 -1 0], [ 0 0 -1], [-1 0 0], [ 0 -1 0], [ 0 1 0] [ 0 0 1] [0 0 1] [ 0 -1 0] [ 0 0 -1] [-1 0 0] [ 1 0 0] [ 0 1 0] [1 0 0] [ 1 0 0] [ 0 1 0] [ 0 0 1] [ 0 0 -1], [-1 0 0], [0 1 0], [ 0 0 1], [ 1 0 0], [ 0 1 0], [ 0 -1 0] [ 0 0 -1] [ 0 0 1] [ 1 0 0] [ 0 0 -1] [ 0 1 0] [-1 0 0] [-1 0 0] [-1 0 0] [ 0 -1 0] [ 0 -1 0] [ 0 0 -1] [ 0 0 -1], [ 0 1 0], [ 0 -1 0], [ 0 0 -1], [-1 0 0], [-1 0 0] ]
- sage.combinat.tiling.ncube_isometry_group_cosets(orientation_preserving=True)#
Return the quotient of the isometry group of the \(n\)-cube by the the isometry group of the rectangular parallelepiped.
INPUT:
n
– positive integer, dimension of the spaceorientation_preserving
– bool (optional, default:True
), whether the orientation is preserved
OUTPUT:
list of cosets, each coset being a sorted list of matrices
EXAMPLES:
sage: from sage.combinat.tiling import ncube_isometry_group_cosets sage: sorted(ncube_isometry_group_cosets(2)) [[ [-1 0] [1 0] [ 0 -1], [0 1] ], [ [ 0 -1] [ 0 1] [ 1 0], [-1 0] ]] sage: sorted(ncube_isometry_group_cosets(2, False)) [[ [-1 0] [-1 0] [ 1 0] [1 0] [ 0 -1], [ 0 1], [ 0 -1], [0 1] ], [ [ 0 -1] [ 0 -1] [ 0 1] [0 1] [-1 0], [ 1 0], [-1 0], [1 0] ]]
sage: sorted(ncube_isometry_group_cosets(3)) [[ [-1 0 0] [-1 0 0] [ 1 0 0] [1 0 0] [ 0 -1 0] [ 0 1 0] [ 0 -1 0] [0 1 0] [ 0 0 1], [ 0 0 -1], [ 0 0 -1], [0 0 1] ], [ [-1 0 0] [-1 0 0] [ 1 0 0] [ 1 0 0] [ 0 0 -1] [ 0 0 1] [ 0 0 -1] [ 0 0 1] [ 0 -1 0], [ 0 1 0], [ 0 1 0], [ 0 -1 0] ], [ [ 0 -1 0] [ 0 -1 0] [ 0 1 0] [ 0 1 0] [-1 0 0] [ 1 0 0] [-1 0 0] [ 1 0 0] [ 0 0 -1], [ 0 0 1], [ 0 0 1], [ 0 0 -1] ], [ [ 0 -1 0] [ 0 -1 0] [ 0 1 0] [0 1 0] [ 0 0 -1] [ 0 0 1] [ 0 0 -1] [0 0 1] [ 1 0 0], [-1 0 0], [-1 0 0], [1 0 0] ], [ [ 0 0 -1] [ 0 0 -1] [ 0 0 1] [0 0 1] [-1 0 0] [ 1 0 0] [-1 0 0] [1 0 0] [ 0 1 0], [ 0 -1 0], [ 0 -1 0], [0 1 0] ], [ [ 0 0 -1] [ 0 0 -1] [ 0 0 1] [ 0 0 1] [ 0 -1 0] [ 0 1 0] [ 0 -1 0] [ 0 1 0] [-1 0 0], [ 1 0 0], [ 1 0 0], [-1 0 0] ]]